Esmeralda wants to solve for x in the equation 3^−2x+1 ⋅3^−2x−3 =3^−3 . Which of the following answers should she select?(1 point)

Question

Responses

x=1/4
x= -5+sqrt73/8
x=-2
x=-4

Answer 1

To solve the equation \( 3^{-2x+1} \cdot 3^{-2x-3} = 3^{-3} \), we can first simplify the left side using the properties of exponents:

\[ 3^{-2x+1} \cdot 3^{-2x-3} = 3^{(-2x+1) + (-2x-3)} = 3^{-4x - 2} \]

Thus, the equation becomes:

\[ 3^{-4x - 2} = 3^{-3} \]

Since the bases are the same, we can set the exponents equal to each other:

\[ -4x - 2 = -3 \]

Now, solve for \( x \):

1. Add 2 to both sides: \[ -4x = -3 + 2 \] \[ -4x = -1 \]

2. Divide by -4: \[ x = \frac{-1}{-4} = \frac{1}{4} \]

So, the solution for \( x \) is:

\[ \boxed{\frac{1}{4}} \] Thus, Esmeralda should select \( \frac{1}{4} \) as her answer.